how to pick up all combination of numbers from multiple vectors

I have a number of vectors, probably with different lengths, e.g., a=[1 2 3], b=[4 5 6 7] and c=[8 9 10 11 12]. From a, b and c I have 3*4*5=60 possible points, e.g., one possibility is (1,4,8). If I know the number of vectors and the length of each vector in advance, this is easy to program. However, I want to write a general code that can find all these combinations regardless of the number of vectors and their individual lengths

[dpb]

The problem is using named variables instead cell array or struct with named fields or somesuch.

If do that, you can simply catenate the vectors programmatically with no problem. Of course, you can easily run into memory problems here in that the number grows geometrically.

[Mike Croucher]

As of R2023a, you can use the combinations function. Details at The new combinations function in MATLAB – for cartesian products and parameter sweeps » The MATLAB Blog - MATLAB & Simulink (mathworks.com)

Answer 1

[John D'Errico]

Don't store your vectors separately. Instead, learn to use tools like cell arrays, which make things hugely more efficient.

V = {[1 2 3],[4 5 6 7],[8 9 10 11 12]};

Next, how do you use a cell array for this purpose? You pass the elements into ndgrid, using what is called acomma separated list. (Or meshgrid.) That is what you get when you type V{:}, a comma separated list. It allows you to pass in each element of the cell array into a function as if each element of the cell array was an argument of the function.

For example, if we did this:

[G1,G2,G3] = ndgrid(V{:});

Hmm. That creates three different arrays, that do contain all combinations of the elements of those vectors if you look carefully. But here we don't want to split the results into n different named arrays. We want a cell array as output. So now, try this:

C = cell(1,numel(V));
[C{:}] = ndgrid(V{:})
C =
  1×3 cell array
    {3×4×5 double}    {3×4×5 double}    {3×4×5 double}

Better. We have captured the output from ndgrid back into a cell array. But what we probably wanted was one flat array, with three columns, and here, 60 rows. We could convert each of those arrays into a column vector easily enough.

C = cellfun(@(X) reshape(X,[],1),C,'UniformOutput',false)
C =
  1×3 cell array
    {60×1 double}    {60×1 double}    {60×1 double}

And, now finally, just convert C into a flat array, using a tool like horzcat. (square brackets will suffice. That is...

C = horzcat(C{:})
C =
     1     4     8
     2     4     8
     3     4     8
     1     5     8
     2     5     8
     3     5     8
     1     6     8
     2     6     8
     3     6     8
     1     7     8
     2     7     8
     3     7     8
     1     4     9
     2     4     9
     3     4     9
     1     5     9
     
     ...
         
     2     7    12
     3     7    12

As you should see, nothing I did was dependent on the size of the arrays, the length of the vectors, the number of different vectors. That was all driven by the one initial cell array as I created it. Learn to use MATLAB, as it was designed to be used.

[mohamed Faraj]

Thank you John. This was helpful. I wonder if you can help with this question

I have a cell array with a variable number of cells and a variable number of elements within each cell. For instance, it could by V{1}{1}=1 2 3 4, V{1}{2}=3 4 1 0,V{1}{3}=3 5 -1 -9, and V{2}{1}=-1 -2 9, V{2}{2}= 0 9 6, i.e., the cell 1 has 3 vectors of length 4 elements and the cell 2 has 2 vectors of length 3. I want to construct another cell array such that it has all combinations of these vectors, here I could have 6 combinations (3*2=6). The new cell array has to be of length 6 cells (number of possible combinations), and each cell has a possible combinations, for example A{1}={1 2 3 4 and -1 -2 9} and A{2}={1 2 3 4 and 0 9 6} . Note that I will get these vectors from a matlab code so I do not know them in advance.

[Afrad Patan]

Hi john, in the same way i need to generate combinations for large number of array elements like If A=1:15, B=1:17, C=1:10, D=1:4, .....X=1:9, Y=1:13 and final array like Z={A,B,C,.....X,Y}. from your solution iam getting out of memory (requesting 107.3 GB for your large no.of array elements, this may slow your matlab unresponsive) error, what can i do here?i have 16gb internal memory in my pc still i need large amount of memory.

[진환 유]

it was really helpful for me. Thank you for good solution!

Answer 2

[Adam Danz]

Here are two solutions for before/after R2023a.

Use combinations in MATLAB R2023a or later

Starting in R2023a, use the combinations function. Details at The new combinations function in MATLAB – for cartesian products and parameter sweeps » The MATLAB Blog - MATLAB & Simulink

% Demo data
a = 1:3; 
b = 11:13; 
c = -4:1; 
d = 9; 

allCombinations = combinations(a,b,c,d)

allCombinations = 54×4 table
    a    b     c     d
    _    __    __    _

    1    11    -4    9
    1    11    -3    9
    1    11    -2    9
    1    11    -1    9
    1    11     0    9
    1    11     1    9
    1    12    -4    9
    1    12    -3    9
    1    12    -2    9
    1    12    -1    9
    1    12     0    9
    1    12     1    9
    1    13    -4    9
    1    13    -3    9
    1    13    -2    9
    1    13    -1    9

Before R2023a

This solution places all vectors in a cell array and uses ndgrid to create permutations. See comments for more detail. T

% Use the same demo data created above.
% Put all vectors into cell array
allVecs = {a,b,c,d}; 

sub = cell(1,numel(allVecs));
[sub{:}] = ndgrid(allVecs{:});
sub = cellfun(@(x)x(:),sub,'UniformOutput', false);

% allPerms is [m x n] matrix of m permutations of n vectors
% m should equal prod(cellfun(@numel,allVecs))
% n should equal numel(allVecs)
allPerms = cell2mat(sub)

allPerms = 54×4

1 11 -4 9 2 11 -4 9 3 11 -4 9 1 12 -4 9 2 12 -4 9 3 12 -4 9 1 13 -4 9 2 13 -4 9 3 13 -4 9 1 11 -3 9

Compare results

combinations() produces a table and my second solution produces a matrix with rows in a different order. Here I'll resort the rows of the table to match the rows of the matrix and we'll confirm that the two solutions are equal.

allCombinationsMat = sortrows(allCombinations{:,:},[4 3 2 1]);
isequal(allCombinationsMat, allPerms)

ans = logical
   1

[mohamed Faraj]

Thank you Adam. The vectors are generated by the code itself. I saved these vectors which I got from matlab in a matrix or a cell aray with a name, e.g., x{1}=1:3, x{2}=b = 11:13, x{3}=-4:1,x{4}=9.

[Adam Danz]

Sounds good. So your variable "x" is my variable "allVecs". Did you have any trouble implementing the rest of the solution?

Answer 3

[Bruno Luong]

a=[1 2 3], b=[4 5 6 7], c=[8 9 10 11 12]

C = {a,b,c}; % put you vectors here

n = length(C);
[C{:}] = ndgrid(C{:});
C = reshape(cat(n+1,C{:}),[],n)

[Mohamad Javadzadeh]

Hi

Tnx for your great anwser

could we limit sum of rows?

[Bruno Luong]

"could we limit sum of rows?"

Simply post filter what you want to keep (such as limit of sum) afterward.

Answer 4

[Chris]

I use allcomb from the Exchange - works great. Works with chars too. It uses ndgrid under the hood and is probably mostly a packaged up version of the code other have shown here.